Definition:Unique

Definition

Suppose $A$ and $B$ are two objects whose definition is in terms of a given set of properties.

If it can be demonstrated that, in order for both $A$ and $B$ to fulfil those properties, it is necessary for $A$ to be equal to $B$, then $A$ (and indeed $B$) is unique.

Equivalently, there is one and only one, or exactly one, such object.

Thus, intuitively, an object is unique if there is one, and only one, of it.

In the language of predicate logic, uniqueness can be defined as follows:

Let $P \left({x}\right)$ be a propositional function and let $x$ and $y$ be objects.

$\exists !x: P \left({x}\right) \dashv \vdash \exists x : \left(P \left({x}\right) \land P \left({y}\right)\right) \iff x=y$

In natural language, this means:

There exists exactly one $x$ with the property $P$

is logically equivalent to:

There exists an $x$ such that $x$ and $y$ have the property $P$ iff $x$ and $y$ are the same object.

Sources

• Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (1936) $\S 4.26$